calculus and nasa michael bloem february 15, 2008 calculus field trip presentation michael bloem...
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Calculus and NASACalculus and NASA
Michael Bloem
February 15, 2008
Calculus Field Trip Presentation
Michael Bloem
February 15, 2008
Calculus Field Trip Presentation
OutlineOutline
NASA’s (many!) uses of calculus– Space– Airfoil design
My use of calculus at NASA– Optimization for air traffic management
NASA’s (many!) uses of calculus– Space– Airfoil design
My use of calculus at NASA– Optimization for air traffic management
€
mgm
€
Ft
€
d2y
dt 2=
(Ft − mgm )
m
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.€
yerr
If then
€
yerr > −(1/2γ y )dyerr
dt
⎛
⎝ ⎜
⎞
⎠ ⎟2
+ y⋅
db
€
Fcontrol = −F∗
Airfoil DesignAirfoil Design
FoilSim Pressure is change in force per area
On a wing, the lift is the difference between the forces acting on the bottom and top of the wing
FoilSim Pressure is change in force per area
On a wing, the lift is the difference between the forces acting on the bottom and top of the wing
€
P(A) =dF
dA
€
P(A)dA∫ = F
€
L = FB − FT = PB (A)dA∫ − PT (A)dA∫L = PB (A) − PT (A)[ ]dA∫
Airfoil Design: Computing LiftAirfoil Design: Computing Lift
Airfoil Design: Computing LiftAirfoil Design: Computing Lift
FoilSim says pressure = 7731 lb How could I improve my estimate?
FoilSim says pressure = 7731 lb How could I improve my estimate?
Rectangle Top [psi] Bottom [psi] Height [psi] Width [in2] Pressure [lb]1 14.80 14.10 0.70 3600 25202 14.75 14.22 0.53 3600 19083 14.75 14.35 0.40 3600 14404 14.72 14.60 0.12 3600 432
TOTAL 6300
Traffic Flow ManagementTraffic Flow Management
Planning of air traffic to avoid exceeding airport and airspace capacity, and effective use of available capacity
Cost of Delay to airlines in 2005 ~ $5.9 Billion (Air Transportation Association Estimate)
Planning of air traffic to avoid exceeding airport and airspace capacity, and effective use of available capacity
Cost of Delay to airlines in 2005 ~ $5.9 Billion (Air Transportation Association Estimate)
3D Visualization of Air Traffic3D Visualization of Air Traffic
Air Traffic Flow ModelsAir Traffic Flow Models
Lagrangian Eulerian
Keep track of each plane Keep track of the numberof planes in different areas
Aggregate Flow ModelAggregate Flow Model
Region i
€
x i(k)
Departures from Center i
€
ud ,i(k)
Inflow from Center j
€
β jix j (k)
Outflow to Center j
€
βij x i(k)
Arrivals into Center i
€
ua,i(k)
€
x(k +1) = A(k)x(k) + ud (k) − ua (k)
€
x i(k +1) = x i(k) − β ij x i(k) − ua,i(k) +j=1j≠ i
N
∑ β jix j (k) +j=1j≠ i
N
∑ ud ,i(k)
Optimization with the Aggregate Flow Model
Optimization with the Aggregate Flow Model
Minimize: quadratic cost on the difference between the scheduled and actual arrivals and departures
Subject to:•Follow system dynamics equations•Do not have more cumulative arrivals or departures than scheduled•Count of aircraft in each center stays below a time-varying maximum•Cumulative arrivals and departures are non-decreasing
Optimization with the Aggregate Flow Model
Optimization with the Aggregate Flow Model
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
How do we optimize?How do we optimize?
Consider a simple case with one variable
Check convexity:
Set derivative = 0:
Consider a simple case with one variable
Check convexity:
Set derivative = 0:€
minimize f (x) = 2x 4 +10x + 5
€
d2 f (x)
dx 2= 24x 2 > 0
€
df (x)
dx= 8x 3 = −10
x =1.0772
Another way to optimize?Another way to optimize?
Newton’s Method– Find where derivative = 0– Iteration:
– Why? Works well on a computer Works well on big problems (many variables)
Newton’s Method– Find where derivative = 0– Iteration:
– Why? Works well on a computer Works well on big problems (many variables)€
xn +1 = xn −′ f (xn )′ ′ f (xn )
Picture for Newton’s MethodPicture for Newton’s Method
Newton’s MethodNewton’s MethodIteration x df(x)/dt d2f(x)/dt 2
1 25 125010 150002 16.666 37042.5928 6666.133343 11.1091665 10978.1762 2961.925954 7.40273485 3255.38758 1315.21165 4.92755323 967.158718 582.7387396 3.26787514 289.181315 256.296197 2.13956607 88.3550679 109.8658318 1.33535727 29.0494487 42.79629689 0.6565731 12.2643275 10.3461177
10 -0.52883073 8.81684939 6.7118865211 -1.8424479 -40.0351994 81.470742412 -1.35104205 -9.72861447 43.807550913 -1.12896585 -1.51151283 30.589533314 -1.0795531 -0.06519094 27.970437715 -1.0772224 -0.00014064 27.849794116 -1.07721735 -6.5934E-10 27.84953317 -1.07721735 0 27.84953318 -1.07721735 0 27.84953319 -1.07721735 0 27.84953320 -1.07721735 0 27.849533
Constrained OptimizationConstrained Optimization
What if we have bounds on x? Optimality condition for a convex function
and a convex constraint set
What if we have bounds on x? Optimality condition for a convex function
and a convex constraint set
€
x is optimal if and only if
df (x)
dx(y − x) ≥ 0
for all y that meet constraints
Example of Constrained Optimization
Example of Constrained Optimization
Constrained optimization problem?
Is it convex? Try our condition
Constrained optimization problem?
Is it convex? Try our condition
€
minimize f (x) = 2x 4 +10x + 5
subject to x ≥ 0
€
x is optimal if and only if
df (x)
dx(y − x) ≥ 0
for all y that meet constraints
€
x is optimal if and only if
(8x 3 +10)(y − x) ≥ 0
for all y ≥ 0
Example of Constrained Optimization (continued)Example of Constrained Optimization (continued)
€
x is optimal if and only if
(8x 3 +10)(y − x) ≥ 0
for all y ≥ 0
€
x is optimal if and only if
(8x 3 +10)y ≥ (8x 3 +10)x
for all y ≥ 0
€
x is optimal if and only if
y ≥ x
for all y ≥ 0
€
x is optimal if and only if
x = 0
Optimal Traffic Flow Management
Optimal Traffic Flow Management
ConclusionsConclusions
NASA uses calculus a lot because calculus helps solve real problems
NASA uses calculus a lot because calculus helps solve real problems
WebsitesWebsites
Altair Lunar Lander CFD at Ames FoilSim Aviation Systems Division
Altair Lunar Lander CFD at Ames FoilSim Aviation Systems Division